We show that immersed Lagrangian Floer cohomology in compact rational
symplectic manifolds is invariant under Maslov flow; this includes coupled mean
curvature/Kähler–Ricci flow in the sense of Smoczyk (Leipzig University, 2001). In
particular, we show invariance when a pair of self-intersection points is born or dies at
a self-tangency, using results of Ekholm, Etnyre and Sullivan (J. Differential
Geom. 71 (2005) 177–305). Using this we prove a lower bound on the time for
which the immersed Floer theory is invariant under the flow, if it exists.
This proves part of a conjecture of Joyce (EMS Surv. Math. Sci. 2 (2015)
1–62).
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